scholarly journals Strong Embeddings for Transitory Queueing Models

2021 ◽  
Author(s):  
Prakash Chakraborty ◽  
Harsha Honnappa
Keyword(s):  
Error Bounds ◽  
Strong Approximation ◽  
Performance Metrics ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Sample Path ◽  
Queueing Models ◽  
Embedding Theorems ◽  
Strong Embedding ◽  
Approximation Theorems

In this paper, we establish strong embedding theorems, in the sense of the Komlós-Major-Tusnády framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes in the form of functional strong approximation theorems.

Relations Among Moments of Waiting Time and Queue Size Distribution in Bulk Queueing Systems

1987 ◽  
Vol 36 (1-2) ◽  
pp. 63-68
Author(s):  
A. Ghosal ◽  
S. Madan ◽  
M.L. Chaudhry
Keyword(s):  
Size Distribution ◽  
Waiting Time ◽  
Queueing Systems ◽  
Queueing Models ◽  
Queue Size ◽  
Different Types

This paper brings out relations among the moments of various orders of the waiting time and the queue size in different types of bulk queueing models.

scholarly journals Concavity of the throughput of tandem queueing systems with finite buffer storage space

1990 ◽  
Vol 22 (03) ◽  
pp. 764-767 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Sample Path ◽  
Concave Function ◽  
Time Interval ◽  
Single Server ◽  
Finite Buffer ◽  
Buffer Storage ◽  
Unlimited Supply ◽  
Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers

1979 ◽  
Vol 11 (02) ◽  
pp. 448-455 ◽  
Author(s):  
David Sonderman
Keyword(s):  
Probability Space ◽  
Queueing Systems ◽  
Stochastic Order ◽  
Sample Path ◽  
System Size ◽  
Waiting Room ◽  
Virtual Waiting Time ◽  
Number Of Customers ◽  
Multi Server ◽  
Quantities Of Interest

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.

scholarly journals Factorization Identities for Reflected Processes, with Applications

2013 ◽  
Vol 50 (03) ◽  
pp. 632-653
Author(s):  
Brian H. Fralix ◽  
Johan S. H. van Leeuwaarden ◽  
Onno J. Boxma
Keyword(s):  
Lévy Processes ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Levy Processes ◽  
Initial State ◽  
Batch Arrivals ◽  
Preemptive Resume ◽  
State Dependent ◽  
Arbitrary Initial State ◽  
And Diffusion

We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth‒collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Lévy case, our identities simplify to both the well-known Wiener‒Hopf factorization, and another interesting factorization of reflected Lévy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth‒death processes and diffusion processes.

Stability of a GI/G/1 Queue: A Survey

2018 ◽  
Vol 35 (03) ◽  
pp. 1850015 ◽  
Author(s):  
Yichi Shen ◽  
Kan Wu
Keyword(s):  
Mathematical Model ◽  
Growth Rate ◽  
Sample Path ◽  
Fundamental Importance ◽  
Queueing Models ◽  
Traffic Intensity ◽  
The Stability ◽  
Underlying Processes

Stability of queues is of fundamental importance in the application of queueing models. To establish the stability of a queue, one has to utilize a mathematical model to describe the evolution of the queue and then defines stability on the model. However, the types of stability are various according to their underlying processes. In this study, we survey the different underlying processes of a [Formula: see text] queue, classify the various types of stability and study the relations among them. Furthermore, from the viewpoint of sample-path, we propose a new result regarding the growth rate of the queue time when the traffic intensity equals 1.

scholarly journals Application of queueing models with abandonment for Call Center congestion analysis

2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Sidney Carlos Ferrari ◽  
Reinaldo Morabito
Keyword(s):  
Call Center ◽  
Call Centers ◽  
Queueing Systems ◽  
Minimum Cost ◽  
Service Organizations ◽  
Analytical Models ◽  
Queueing Models ◽  
Phone Calls ◽  
Using Data ◽  
Mixed Distributions

Abstract This paper studies and applies queueing systems to Call Centers regarding the possibility of customer abandonment from the system before being served due to their impatience in waiting for a service. Call Centers are service organizations that predominantly serve customers via phone calls. One of the main concerns in managing them is to provide quality service at a minimum cost. Noticing the quality of services offered is expressed by customers, for example by abandonment from the queue. This paper shows that the M/M/c+G analytical queueing models with abandonment, with patience time represented by generic distributions (particularly mixed distributions), are more effective than the M/M/c+M analytical queueing models with abandonment, with Exponential patience, commonly used to evaluate congestion problems in Call Centers and support sizing and operational decisions in these systems. We conducted a study using data extracted from a Bank Call Center located in Israel and the parameters and some performance measures are determined based on this data. These sampling measures are compared with the same measures achieved by the M/M/c+M and M/M/c+G analytical queueing models considered in this research, which use parameters obtained empirically and the mixed and non-mixed distributions based on Exponential and Lognormal to represent user patience. An experimental discrete simulation model was also used to explore an alternative scenario, showing the potential of using the approaches based on analytical models with abandonment for Call Center analysis.

Sample Path Criteria for Weak Majorization

1994 ◽  
Vol 26 (01) ◽  
pp. 155-171 ◽  
Author(s):  
Panayotis D. Sparaggis ◽  
Don Towsley ◽  
Christos G. Cassandras
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Sample Path ◽  
Cumulative Number ◽  
Stochastic Orderings ◽  
Optimal Policies ◽  
Weak Majorization ◽  
Queue Lengths ◽  
And Performance

We present two forms of weak majorization, namely, very weak majorization and p-weak majorization that can be used as sample path criteria in the analysis of queueing systems. We demonstrate how these two criteria can be used in making comparisons among the joint queue lengths of queueing systems with blocking and/or multiple classes, by capturing an interesting interaction between state and performance descriptors. As a result, stochastic orderings on performance measures such as the cumulative number of losses can be derived. We describe applications that involve the determination of optimal policies in the context of load-balancing and scheduling.

Sign in / Sign up

Export Citation Format

Share Document